The arguments about legacy for the Olympic Games tend to dwell on bricks and mortar – or steel and fibreglass. What will we do with the basketball arena? Will West Ham really play their home matches in the Olympic stadium? Can the £9bn capital investment show a concrete return in transport networks and affordable housing?

Infrastructure isn't only about rail links and building projects though. It is also, more fundamentally, about connecting individuals with the wider society around them, building bridges between communities and families, particularly those that feel themselves isolated or hard to reach. That kind of connectivity is harder to measure. Can the ties that bind us be strengthened by an event like this? That was certainly part of the rhetoric that allowed the original bid to be successful and you would have to say that over the past week the country has felt a little more cohesive, a little more purposeful, a little more Team GB. But will any of that last beyond the end of the Games?

One of the most striking aspects of this Olympiad has been the success of the volunteering programme. No visitor to the park and arenas will have come away without having felt genuinely and warmly welcomed (every 20 yards or so). The volunteers in their purple and red uniforms have been the human face of the 2012 Games and of Britain, and a particularly smiley and effervescent one; their pride at being involved, even often in quite a mundane way, has been infectious.

Through active programmes such as Personal Best, which sought to involve those "furthest from work" in the Games, and more widely through the committed engagement of London boroughs, notably Newham in which the Games park is situated, the 70,000 strong army of helpers also begins to reflect all sectors of society both locally and nationally. Over the last seven years, the catalyst of the Games has generated a volunteer programme that is now 8,000 strong in Newham alone. That group has been involved not just in preparing for the Games, but in supporting the activities of the council across the full range of its community involvement, from environmental work to help with children's literacy in library reading groups.

Newham is London's poorest borough and also its most diverse. The enthusiasm of the Newham volunteers has shown, in a small way, not only that everyone can have a part to play but also that the spirit of altruism, and joining in, has the capacity to get to places that policy initiatives find hard to reach.

It was something of this spirit that the prime minister was hoping to foster with his big society idea, without ever demonstrating that he really had the first clue about how to engage the kinds of people who might most benefit from involvement in ideas bigger than themselves. It is one thing to talk about a participatory society; it is another to create the space and structures to help that to come about.

Newham started with 300 volunteers seven years ago. The programme, which is core funded by the council, has spread largely through example and word of mouth, a slow process, but a rewarding one for all of those involved and all those who have benefited from their commitment of time.

The motto of the Games is "Inspire a generation". Narrowly, we might take this as a call to sporting arms, a challenge to find among our children the next Chris Hoy or Jessica Ennis. We can't help but be in awe of the kind of dedication and self-sacrifice that allows our medal winners to have their moments of triumph.

The other smaller acts of sacrifice that the Olympics have involved have been inspiring on their way too, however. The young volunteers down from Scotland or up from Cornwall who have slept in tents or on a mate's floor in order to get up at five or six in the morning and make sure everyone gets a cheery hello in a bus queue or to man a door for a day. The local people, many of them retired, in Newham and beyond, who saw this as an opportunity to tell the world a little bit of what they were about.

One real test of these Olympics is whether that spirit of inclusion, and of wanting to be included, can extend beyond the Games. Big societies do not come about by themselves, they need practical commitment, a real understanding of communities. And they need political inspiration that lasts beyond a fortnight.